PHYSICAL REVIEW B 87, 245134 (2013)

Ballistic charge transport in graphene and light propagation in periodic dielectric structureswith metamaterials: A comparative study

Yury P. Bliokh,1,2 Valentin Freilikher,1,3 and Franco Nori1,4

1CEMS, RIKEN, Saitama, 351-0198, Japan2Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel3Department of Physics, Jack and Pearl Resnick Institute, Bar-Ilan University, Israel

4Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA(Received 26 March 2013; revised manuscript received 14 June 2013; published 28 June 2013)

We explore the optical properties of periodic layered media containing left-handed metamaterials. This studyis based on several analogies between the propagation of light in metamaterials and charge transport in graphene.We derive the conditions when these two problems become equivalent, i.e., the equations and the boundaryconditions for the corresponding wave functions coincide. We show that the photonic band-gap structure of aperiodic system built of alternating left- and right-handed dielectric slabs contains conical singularities similarto the Dirac points in the energy spectrum of charged quasiparticles in graphene. Such singularities in the zonestructure of the infinite systems give rise to rather unusual properties of light transport in finite samples. Inan insightful numerical experiment (the propagation of a Gaussian beam through a mixed stack of normal andmetadielectrics), we simultaneously demonstrate four Dirac point-induced anomalies: (i) diffusionlike decay ofthe intensity at forbidden frequencies, (ii) focusing and defocussing of the beam, (iii) absence of the transverseshift of the beam, and (iv) a spatial analogue of the Zitterbewegung effect. All of these phenomena take placein media with nonzero average refractive index and can be tuned by changing either the geometrical andelectromagnetic parameters of the sample or the frequency and the polarization of light.

DOI: 10.1103/PhysRevB.87.245134 PACS number(s): 81.05.Xj, 72.80.Vp, 03.50.De

I. INTRODUCTION

Highly unusual properties of monolayers of graphite(graphene) and of optical media with negative refractiveindices (left-handed metamaterials) had been independentlypredicted and studied theoretically a long time ago.1,2 Atthat time, however, these predictions were perceived as ratherintriguing but unrealistic exotica and remained unnoticedfor about a half-century, until quite recently (and nearlysimultaneously), they were embodied in real materials. Thisimmediately triggered an explosion of interest and activities,in metamaterials and graphene, both in solid state physics andoptics. Researchers also realized that the most unusual proper-ties of electron transport in graphene were also peculiar to thepropagation of light in dielectric systems with metamaterials.Mathematically, this is because, under some (rather general)conditions, the Maxwell equations for electromagnetic wavesin an inhomogeneous dielectric medium can be reduced to theDirac equations for charge carriers in graphene subjected toan external electric potential.

A. Similarities between Maxwell and Dirac equations

The history of recasting Maxwell equations in alternative,more compact, spinor forms goes back to the beginning ofthe past century, and is still in progress (for a comprehensivehistorical overview see Refs. 3 and 4, with recent examplesin Refs. 5–8). Therefore it is not surprising that the similaritybetween Maxwell and Dirac equations has long been noticed(according to Ref. 3, Majorana discussed it already in 1930). Inthe general case of inhomogeneous media, the close analogybetween (i) the quantum-mechanical form of the equationsfor the Reimann-Silberstain vector fields (linear combinationsof the electromagnetic vectors �D and �B) and (ii) the Dirac

equation, written in the chiral representation of the Diracmatrices, was explicitly demonstrated in Ref. 3.

Recently, as graphene became increasingly more popular insolid state physics, the mathematically established similarityof Maxwell and Dirac equations took on a new physicalsignificance. Inspired by the very unusual predictions anddiscoveries made in graphene, research groups in opticsstarted endeavors to reproduce the unique transport propertiesof graphene in specifically designed dielectric structures.An additional incentive to these efforts came from the fact that,while the elementary building blocks of graphene are fixed,modern micro and nanotechnologies enable manufacturingperiodic dielectric samples with a variety of types and sizesof unit cells. Moreover, the electrodynamic parameters ofphotonic crystals can, in principle, be controlled by externalfields, providing unique opportunities to study condensedmatter phenomena in optical ways; for example, by openinga gap between Dirac cones, as well as breaking and restoringspace-inversion and time-reversal symmetries.5 Rather simpleelectrodynamical analogies furnish physical insights intoproperties and applications of graphene such as the Kleinphenomenon,9 breaking the valley degeneracy,10 graphenequantum dots,11,12 the electronic Goos-Hanchen shift,13 delo-calization in one-dimensional disordered systems,14 etc. Fur-thermore, some exotic phenomena predicted and discovered inoptical systems (like, for example, the “light wheel” localizedmode,15 or confined cavity modes and ministop bands inbroad periodic photonic waveguides16,17) could prompt uniquepossibilities for creating new graphene-based devices. In thisregard, particularly promising is the analogy between photoniccrystal broad waveguides and zigzag graphene nanoribbonsstudied in Ref. 18, where it was shown that the photonicmode coupling also arises in nanoribbons at low energies.

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YURY P. BLIOKH, VALENTIN FREILIKHER, AND FRANCO NORI PHYSICAL REVIEW B 87, 245134 (2013)

FIG. 1. (Color online) Surface ω(kx,ky) described by the disper-sion equation (1) for electromagnetic waves. The contact of two coneslooks like a Dirac point in graphene; however, in contrast to graphene,for light, both cones correspond to the same field.

The implementation of the analogy between Dirac electronsand light could be rewarding for the optical communityas well, because it is relatively easy to create in graphenean inhomogeneous potential pattern with any distributionof p-n and n-n junctions, while designing a periodic orrandom stack of alternating positive-negative dielectric layersis nowadays a feasible task. Thus graphene could provideanalogue laboratory models in order to test the optics ofmetamaterials in a controlled way.

B. Dirac point

The key feature of graphene, from which all its uniquetransport properties stem, is the existence of Dirac cones inthe band structure of its energy spectrum. At first glance,one does not have to work hard to obtain in optics a double-conical, graphenelike dispersion law: it is inherent in any planemonochromatic wave propagating in a homogeneous medium,as the relation between its frequency ω and the wave numberk is given by

ω2 = c2k2. (1)

However, the contact of two cones in Fig. 1 only looks likea Dirac point (DP). In fact, of the two cones in Fig. 1, only one(for example, the upper one) is related to a photon, while thesecond solution of Eq. (1) (lower cone in Fig. 1) is redundant.This lower cone does not carry any additional information andis not related to any different physical entity like, for example,a hole in graphene or a positron in the case of relativisticQED. In other words, the photon and antiphoton are identical.3

Hence the challenge in optics is to create a structure with areal DP in its spectrum, so that different types of waves wouldcorrespond to two different cones (a sort of optical “particle-antiparticle” pair). Appropriate for this purpose are photoniccrystals in which two modes degenerated in a homogeneousspace become split by the periodicity.19

The analytical and numerical studies of two-dimensionalperiodic structures (infinite rods embedded in a background

medium with a different dielectric constant) were carried outas early as in 1991, for square20 and triangular21 lattices. Linearsingularities (that nowadays are called Dirac points) are clearlyseen in the band structures of both systems, although theyescaped the attention of Refs. 20 and 21 mostly interestedin absolute band gaps for different polarizations. Afterwards,during more than two decades, studies of two-dimensionalphotonic crystals were primarily aimed on maximizing thephotonic band gap22–24 (see also the review Ref. 25, andreferences therein), until the discovery of the unusual trans-port properties in graphene, and the potential to reproducethem in optics switched efforts towards the search andfurther exploration of photonic structures with Dirac-conelikesingularities in their transmission spectra.5,7,19,26 A numberof new optical phenomena arising due to the existence ofDirac points were predicted and discovered: diffusionlike1/L-dependence of the pulse intensity on the distance L ofpropagation inside the photonic crystal19,27 (which is unusualfor nonrandom media), oscillatory motion of a Gaussian beam(optical analogue of the Zitterbewegung effect),28–30 extinctionof coherent backscattering,31–33 conical diffraction,34 as wellas the existence of graphenelike and novel edge states.35,36

From the above-mentioned publications, one can concludethat the existence of Dirac points in two-dimensional periodicstructures is a rather universal phenomenon, in the sense thatthey appear irrespective of sample details, such as the shapeand dielectric parameters of the “atoms” and its structuralsymmetry. For example, in the band structures presented inRefs. 7,20,21, and 37, DPs show up at square, triangular, andhoneycomb lattices. The general criteria for the existence ofDP in periodic dielectric samples were discussed in Ref. 38.

The situation in layered periodic media is quite different andless studied. As we show below, DPs cannot exist in periodi-cally layered dielectric structures built of monotype (i.e., witheither all positive or negative refractive indices) dielectrics, nomatter their period, size, and dielectric contrast between thelayers. In Refs. 38 and 39, a one-dimensional periodic arrayof metallic unit cells of a special shape was considered, whichexhibited Dirac points created by the accidental degeneracy oftwo modes. To create photonic Dirac cones in one dimension,both normal and metamaterials should be used. Interestinglyenough, a single DP can exist in a homogeneous dispersivemetamedium at a frequency at which both the dielectricpermittivity and the magnetic permeability approach zerosimultaneously.6 Eigenwaves in structured, infinite periodicsystems built of alternating normal and left-handed dielectriclayers, and the transmission and reflection from finite samples,were analyzed in Refs. 40–42. It was shown that, when the spa-tial average of the refractive index over the period was zero, theband structure consisted of gaps at all frequencies except fora set of isolated points (discrete modes), for which the opticalthicknesses of the adjacent layers were equal to the same inte-ger number of half-wavelengths. New Dirac cones in graphenesupelatices created by double-periodic and quasiperiodicelectrostatic potentials have been considered in Ref. 43.

C. Brief summary

Here we study the transport properties of layered periodicdielectric systems “electronically similar” to graphene, in the

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sense that they possess Dirac cones in their photonic bandgap structures. In Sec. II, we demonstrate, using a simpleexample, and discuss the similarity and differences betweenthe Maxwell and Dirac equations as well as between thecorresponding boundary conditions. Section III presents thetransmission through potential barriers created in grapheneby applying steplike electrostatic potentials, in comparisonwith light propagation in slabs of normal dielectrics andmetamaterials. In Sec. IV, the photonic band-gap structures ofperiodically layered dielectrics are compared with the structureof the electron energy zones of graphene subjected to a periodicpotential. It is shown that, unlike a two-dimensional photoniccrystal, Dirac cones in a layered periodic medium can existonly when the medium consists of alternating slabs of left-and right-handed dielectrics (mixed samples). In Sec. V, westudy the propagation of Gaussian beams of light throughperiodically layered mixed samples built of alternating slabswith positive and negative refractive indices, and of beamsof charge carriers through finite graphene superlattices. Wedemonstrate the anomalous, diffusionlike dependence of theintensity on the propagation distance, and an analog of theZitterbewegung effect in a wide range of parameters. Newunusual transport properties of such samples are predicted. Inparticular, it is shown that two contacting Dirac cones manifestthemselves differently: given two beams with frequenciesbelonging to different cones, one is focused and anotheris defocused. The magnitude of the shift of the focus isindependent on the distance of the sample from the focal planeof the incident beam and is proportional to the width of thesample. At oblique incidence, a Gaussian beam is not displacedalong the sample even at nonzero values of the mean value ofthe dielectric constant.

II. EQUATIONS AND BOUNDARY CONDITIONS

The dynamics of the charge carriers in an external potentialu in graphene is described by a spinor

ψ = (ψA,ψB)T

whose components are related to two sublattices in the unitcell of the crystal.44 In the low-energy limit, near the Diracpoint, the components of this spinor obey the Dirac equations.When the energy w of the charge carrier is fixed, then the timedependence of the spinor is given by exp(−iwt/h), and theseequations can be written as

ψA = − ivFh

w − u(x,y)

(∂ψB

∂x− i

∂ψB

∂y

),

(2)

ψB = − ivFh

w − u(x,y)

(∂ψA

∂x+ i

∂ψA

∂y

),

where u(x,y) is the electrostatic potential and vF is the Fermivelocity.

In order to link this to Maxwell equations, we now consider,as an example, a TE electromagnetic wave (where the magneticfield has only one nonzero z component) propagating ina homogeneous medium and introduce two complex-valuedfunctions

E = Ey − iEx, H = ZHz, (3)

where Z = μ/ε is the medium impedance, μ and ε arethe medium permeability and permittivity, accordingly. InRef. 3, instead of Eq. (3), the Reimann-Silberstain vectorwave functions were used to derive a quantum-mechanicalmatrix form of the classical wave equations in the general caseof arbitrary electromagnetic fields propagating in media withspace-dependent permittivity and permeability. The analogywith the relativistic Dirac equations was noted.

It is easy to show that for monochromatic fields E and H[the time dependence is given by exp(−iωt)], the Maxwellequations yield

H = − i

k0n

(∂E∂x

− i∂E∂y

), E = − i

k0n

(∂H∂x

+ i∂H∂y

).

(4)

Here, n is the medium refractive index, and k0 = ω/c. It isevident that after the replacement

E ↔ ψA, H ↔ ψB, nω ↔ (w − u)/h, c ↔ vF , (5)

Eq. (4) coincides with Eq. (2). Namely, the 2D Maxwellequations for the complex effective fields (3) in a homogeneousmedium and the Dirac equations for the wave functions ofthe charge carriers in graphene become identical. The role ofthe refractive index of the corresponding effective medium isplayed by the quantity neff = (w − u)/hω. Therefore, if, forexample, the potential is a piecewise-constant function of onecoordinate, the corresponding graphene superlattice models alayered dielectric structure.14 In particular, a layer, in which thepotential u exceeds the energy w of the particle, w − u < 0, issimilar to a slab with negative refractive index n. This meansthat a junction of two regions having opposite signs of w − u issimilar to an interface between left- and right-handed dielectricmedia with the refractive indices n1 and n2, if

n1

n2= w − u1

w − u2. (6)

Because of this similarity, a p-n junction can focus Diracelectrons in graphene45 in the same way as the focusing ofelectromagnetic waves by the boundary between a normaldielectric and a metamaterial.1,46 However, it is importantto realize that, as it follows from Eq. (6), a change of w

necessarily implies the corresponding change of the ration1/n2. Thus, to model the same (i.e., with fixed values of u1

and u2) graphene bilayer structure, but at different energies,one has to chose different pairs of dielectrics.

It follows from Eq. (2) that the energy spectrum of thecharge carriers in graphene in a homogeneous potential u =const is linear near k = 0, i.e., consists of two cones touchingat the Dirac point (see Fig. 1):(

w − u

hvF

)2

= k2x + k2

y. (7)

After substituting Eq. (5), Eq. (7) looks exactly like thedispersion law (1) for photons. Whilst Eqs. (2) and (4) are akin,the similarity between the two problems is not complete. First,unlike Dirac wave functions, the genuine electromagneticfields �E and �H are real, i.e., equal to their complex conjugates.Due to this, positive and negative frequencies (upper and lowercones in Fig. 1) correspond to the same fields, in contrast to

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a Dirac spinor, which describes electrons, when (w − u) > 0,and holes, when (w − u) < 0. One further distinction is thatthe electric field �E satisfies the continuity condition

div �E = ∂Ex

∂x+ ∂Ey

∂y= 0 , (8)

which is not required for the Dirac wave functions.Essentially different are also the boundary conditions for the

Dirac wave functions at the interface between two half-spaceswith potentials u1 and u2,

ψ1A = ψ2A, ψ1B = ψ2B, (9)

and for the effective electromagnetic fields E and H at theboundary between two dielectrics,

Z2H1 = Z1H2, Re E1 = Re E2. (10)

Equations (10) follow from the boundary conditions for the realfields, E1y = E2y and H1z = H2z (the boundary is assumed tobe parallel to the y axis).

Although the charge transport in graphene and the lightpropagation in dielectrics are governed by similar equa-tions, Eqs. (2) and (4), inside the medium, the boundaryconditions (9) and (10) are distinct from each other. Thismeans that, generally speaking, the coupling between adjacentsamples in these two cases is different. This difference is moreconspicuous in the particular case of a monochromatic planewave, H ∝ E ∝ exp[i(kxx + kyy − ωt)], when the boundaryconditions for the effective fields take the form

Z2H1 = Z1H2, E(1 + iky/k2x) = E2(1 + iky/k1x), (11)

where kjx =√

k20n

2j − k2

y . It is easy to see that Eqs. (9) and (11)coincide only when ky = 0 (normal incidence) and Z1 = Z2.That is, in this particular instance, the transmission of Diracelectrons through a junction is similar to the transmission oflight through an interface between two media with differentrefractive indices but equal impedances. In other words, atnormal incidence, any junction in graphene, either n-n, p-p,or p-n, is analogous to a contact between two perfectlymatched dielectrics or microwave elements. Such an interfaceis absolutely transparent to the normally incident radiationand therefore to the Dirac electrons in graphene as well.This provides a more intuitive insight into the physics of theKlein paradox (perfect transmission through a high potentialbarrier44,47) in graphene systems.

III. TRANSMISSION THROUGH POTENTIAL BARRIERSAND DIELECTRIC SLABS: SIMILARITIES

AND DIFFERENCES

To compare the transport properties of Dirac electrons andlight in the general case of oblique incidence, ky �= 0, wefirst consider the transmission of particles through a steplikepotential barrier, i.e., through the line x = 0 separating twodomains (1 and 2) of a graphene sheet with different valuesu1 and u2 of the potential. In what follows, we will consideronly one spinor component, say ψA ≡ ψ , because ψB could befound using Eq. (2). The solutions of Eq. (2) in both domainscan be presented as a linear combination of plane waves with

equal (at the chosen geometry of the system) values of ky :

ψj = ei[kyy−(w−uj )t/h][ψ (+)j eikjxx + ψ

(−)j e−ikjxx], j = 1,2.

(12)

From now on, we consider the range of parameters where thereare no total internal reflections and therefore Imkx = 0. Fromthe continuity of the wave functions at the boundary, Eq. (9),it follows (the incident wave now propagates from medium 1to medium 2) (

ψ(+)2

ψ(−)2

)= M1→2

(ψ

(+)1

ψ(−)1

), (13)

where the transfer matrix M1→2 for the interface x = 0 is equalto

M1→2 = 1

2 cos θ2

∣∣∣∣∣∣∣∣∣∣ g

(+)1→2 g

(−)1→2

g(−)1→2

∗g

(+)1→2

∗

∣∣∣∣∣∣∣∣∣∣ (14)

and the matrix elements are

g(±)1→2 = e−iθ2 ± s1s2e

±iθ1 . (15)

Here, sj = sgn(w − uj ), θ1 and θ2 are the angles of incidenceand refraction, respectively, while θj = arctan(ky/kjx).

We determine the transmission T and reflection R coeffi-cients as the ratios of the normal-to-the-boundary componentsof the densities of the transmitted, �J (+)

2 , and reflected, �J (−)1 ,

currents divided by the normal component of the incidentcurrent density J

(+)1x :

T = J(+)2x /J

(+)1x , R = J

(−)1x /J

(+)1x , (16)

where

J(±)jx = ±2sj cos θj |ψ (±)

j |2. (17)

From Eqs. (16) and (17), the following formulas can beobtained (see, e.g., Refs. 48 and 49):

T = 2 cos θ1 cos θ2

1 + cos(s1θ1 + s2θ2), R = 1 − cos (s1θ1 − s2θ2)

1 + cos(s1θ1 + s2θ2),

(18)

where the angles of incidence θ1 and refraction θ2 areconnected by the relation

sin θ2 = w − u1

w − u2sin θ1. (19)

The same signs of s1 and s2 correspond to n-n (or p-p)junctions, while for an n-p (or p-n) junction s1 = −s2. It iseasy to see that at normal incidence (θ1 = θ2 = 0) the potentialbarrier of any height is absolutely transparent at any energy(the Klein tunneling effect).

In the case of light propagating through the interfacebetween two dielectrics, whose parameters are ε1, μ1 and ε2,μ2, the transfer matrix

M1→2 = 1

2 cos θ2

∥∥∥∥∥ G(+)1→2 G(−)

1→2

G(−)1→2

∗ G(+)1→2

∗

∥∥∥∥∥ (20)

connects the amplitudes of leftward and rightward propagatingwaves at both sides, and has the same form as Eq. (14), with

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BALLISTIC CHARGE TRANSPORT IN GRAPHENE AND . . . PHYSICAL REVIEW B 87, 245134 (2013)

the matrix elements g(±)2→1 replaced by

G(±)1→2 = cos θ2 ± s1s2 cos θ1

Z2

Z1(21)

for TE waves and by

G(±)1→2 = cos θ2

Z2

Z1± s1s2 cos θ1 (22)

for TM radiation. In Eqs. (21) and (22), sj = sgn nj , nj =±√

εjμj , and Zj = √μj/εj . For left-handed dielectrics (ε <

0, μ < 0), the refractive index n is negative.When Z1 = Z2, the expressions (21) and (22) are identical

and are related to the matrix elements (15) of the correspondingtransfer matrix in graphene, Eq. (13), as

G(±)1→2 = Re g

(±)1→2. (23)

This relation is a consequence of the above mentioneddifference between the solutions of the Dirac and Maxwellequations: the former are complex-valued functions, while theelectromagnetic fields are real.

The light transmission and reflection coefficients are deter-mined as the ratios of the normal components of the transmitted(for T ) and reflected (for R) energy fluxes divided by thenormal component of the incident energy flux. When Z1 = Z2

(the situation most favorable for the analogy between Diracelectrons and light), T and R take the forms

T = 4 cos θ1 cos θ2

(cos θ1 + cos θ2)2 , R = (cos θ1 − cos θ2)2

(cos θ1 + cos θ2)2 . (24)

One can see that even in the particular case, Z1 = Z2, Eqs. (18)and (24) are, generally speaking, different, and coincide onlywhen θ1 = θ2 = 0, i.e., when the boundary conditions (11) areequivalent. This means that in spite of the identity of Eqs. (2)and (4), the analogy between the transport of Dirac electrons ingraphene and electromagnetic radiation in dielectrics shouldnot be extended too far. Due to the differences in the boundaryconditions, the analogy holds only for normal incidence on theinterface between two perfectly matched media.

The transmission of Dirac electrons trough a potentialbarrier of finite width d, and of an electromagnetic wavethrough a dielectric slab of the same width are described bysimilar matrices of the form

B = A2→1Sd A1→2, (25)

where the indices 1 and 2 now correspond, respectively, to theoutside and inside of the barrier (slab). The matrix A is equalto M , as in Eq. (14) for graphene, and A = M, as in Eq. (20)for dielectrics. Obviously, A2→1A1→2 = I , where I is a unitmatrix. The diagonal matrix Sd = diag(eiϕ,e−iϕ), with ϕ =k2xd describes the propagation inside the barrier (dielectricslab). From Eq. (25), it follows that under the condition

k2xd = mπ, m = 1,2,3, . . . , (26)

B = (−1)mI . This means that either a potential barrier or alayer of dielectric is transparent if its width is equal to aninteger number of half-wavelengths. This is a general propertyof both, Dirac and Maxwell equations, which is independentof the ratio between the impedances, and its physical naturehas nothing to do with the Klein tunneling.

IV. PERIODICALLY STRIPPED GRAPHENESUPERLATTICES AND PHOTONIC STRUCTURES

In this section, we compare the structure of the electronenergy zones of graphene subject to a periodic electrostaticpotential, with the photonic band gap structures of periodicallylayered dielectrics. To do this, we assume that the potential ingraphene takes the two values u1 and u2 in alternating areasof widths d1 and d2, and the corresponding dielectric sampleis built of alternating layers of the same thicknesses, d1 andd2, with refractive indices n1 and n2, and impedances Z1 andZ2, respectively. In both cases, the propagation in the layers1 and 2 is described by the matrices Sj = diag(eiϕj ,e−iϕj ),j = 1,2, where ϕj = djkjx . Assuming that the layers areparallel to the y axis, the transformation matrix P on theperiod D = d1 + d2 is defined by ψ(x + D) = Pψ(x) and isequal to

P = S1M2→1S2M1→2 (27)

for graphene and

P = S1M2→1S2M1→2 (28)

for dielectrics.The eigenvalues λ = exp(ik‖D) (k‖ is the Bloch wave

number along the x axis) of the matrix P (P) depend onthe energy (frequency) and on the tangent component ky of thewave vector. Both periodic structures are transparent if

|λ| = 1. (29)

In other words, Eq. (29) determines the transparency zones:the ranges of the energies (frequencies) and wave numbers ky

for which the longitudinal wave number k‖ is real.50 It can beshown that

λ = F ±√

F 2 − 1, (30)

where

F (w,ky) = cos ϕ1 cos ϕ2

+(

tan θ1 tan θ2 − s1s2

cos θ1 cos θ2

)sin ϕ1 sin ϕ2

(31)

for Dirac electrons in graphene and

F (ω,ky) = cos ϕ1 cos ϕ2

− s1s2

2

(Z1 cos θ1

Z2 cos θ2+ Z2 cos θ2

Z1 cos θ1

)sin ϕ1 sin ϕ2

(32)

for electromagnetic waves in layered dielectrics. In both cases,the eigenmodes obey the dispersion equation

cos(k‖D) = F. (33)

It follows from Eqs. (29) and (30) that a periodic structureis transparent for the points in the plane (ky,w) [or (ky,ω)] forwhich the inequality |F | � 1 holds. On the other hand, in eachof these two planes, the conditions

ϕ1 = pπ, p = 1,2,3, . . . ,(34)

ϕ2 = qπ, q = 1,2,3, . . .

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YURY P. BLIOKH, VALENTIN FREILIKHER, AND FRANCO NORI PHYSICAL REVIEW B 87, 245134 (2013)

(p and q are positive integer numbers) determine two setsof curves, where P and P are equal to (−1)p+q I . Thismeans that those types of curves belong to transparencyzones. It is apparent that since at the crossings of these linesthe eigenvalues of the matrices P and P are equal to ±1,such singular crossing points lie at the edges of the zonesand represent their points of contact, known as “conicalintersection,” or “diabolic points” (see, e.g., Ref. 51).

The coordinates, kyt and k0t = ωt/c, of these points in theplane (ky,k0) can be found from Eq. (34):

k20t = [(pπ/d1)2 − (qπ/d2)2]

/(n2

1 − n22

),

(35)k2yt = [

n22(pπ/d1)2 − n2

1(qπ/d2)2]/(

n21 − n2

2

).

In the case of graphene, the coordinates of the diabolic pointsin the plane (w,ky) are calculated in the same way. In thevicinity of the points defined by Eq. ( (34)), the phases ϕ1 andϕ2 can be written as

ϕ1 = pπ + δϕ1, ϕ2 = qπ + δϕ2, (36)

where |δϕj | � 1. Substituting Eq. (36) into Eq. (32) yields

F � (−1)p+q[1 − 1

2

(δϕ2

1 + δϕ22 + 2aδϕ1δϕ2

)]≡ [

1 − 12f (δϕ1,δϕ2)

](37)

with

a = s1s2

2

(Z1 cos θ1

Z2 cos θ2+ Z2 cos θ2

Z1 cos θ1

)

≡ 1

2

(k1xε2

k2xε1+ k2xε1

k1xε2

),

where the components k1x and k2x are taken at the point(kyt ,k0t ). In the plane (δky,δk0), where δk0 and δky are smalldeviations of k0 and ky from their values given by Eq. (35),the points for which the quadratic form f (δϕ1,δϕ2) (37) ispositive, constitute a transparency zone, while the points withf < 0 correspond to a gap in the photonic spectrum.

The relation between δϕj , δky , and δk0 follows from theformula

ϕj = dj

√n2

j (k0)k20 − k2

y, (38)

where nj (k0) is the refraction index of the j th layer. Ananalogue formula for graphene has the form

ϕj = dj

√[(w − uj )/hvF ]2 − k2

y. (39)

While in conventional dielectrics, the dispersion can beignored, if it is weak enough, in left-handed materials, it mustalways be taken into account. Indeed, the surface ω(k), wherek =

√k2x + k2

y , for normal dielectrics, is a cone similar to theone presented in Fig. 1; i.e., ω(k) = n−1ck, with n = const >

0. For left-handed metamaterials n < 0, and the group velocityvg = (dω/dk) is negative, vg < 0, i.e., is antiparallel to thephase velocity ω/k. The surface ω(k), in a small vicinity of acertain frequency ω0, has the form depicted in Fig. 2. Althoughlocally it looks like a part of the lower cone in Fig. 1, ω(k) �=n−1ck, with n = const on this surface, because the contactpoint of the cones is shifted from the origin.

Although the absolute values of the transmission andreflection coefficients in Eqs. (18) and (24) are different, the

FIG. 2. (Color online) Surface ω(kx,ky) in a small vicinity of acertain frequency for left-handed media.

dispersion characteristics of two adjoining right- and left-handed dielectric layers, and the energetic spectrum diagramsof n-p junction in graphene are identical, as it is shownschematically in Fig. 3. As a consequence of this identity, theperiodic dielectric structure and graphene superlattice formedby a periodic external potential possess the same uniquetransport properties.

A. Photonic structure

From Eq. (38), we obtain

δϕj = dj

kjx

(ckjt

vjg

δk0 − kyt δky

), (40)

where vjg is the wave group velocity in the j th layer at thefrequency ωt = ck0t . The group velocity in Eq. (40) is positive

FIG. 3. (Color online) The dispersion characteristics ω(k), shownas blue lines, of two adjoined right-handed (left panel) and left-handed(right panel) dielectric layers is identical to the energetic spectrumw(ky) of n-p junction in graphene. The horizontal dashed red linerepresents the wave frequency, or the quasiparticle energy. The cones’apexes correspond to the zones touching points.

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for normal dielectric layers, and negative for layers of left-handed metamaterials.

Equation (40) shows that the phase variations δϕj vanishon the lines

δk0 = vjgkyt

ckjt

δky. (41)

These lines lie in the transparency zones and intersect atthe point where they touch. The line associated with theright-handed dielectric lies in the first and the third quadrants,whereas the line associated with the left-handed metamateriallies in the second and the fourth quadrants, as it is shown inFig. 4. When the lattice is composed of conventional dielectricsonly, both lines δϕ1(δky,δk0) = 0 and δϕ2(δky,δk0) = 0 lie inthe first and the third quadrants, and the transparency zonestouch one another in a manner shown in Fig. 4(a). There areband gaps above and below the frequency ωt = ck0t , and these

FIG. 4. (Color online) Transparency zones (gray) in the planeω(kx,ky) for (a) a lattice composed of conventional dielectrics only;(b) and (c) lattices containing both left- and right-handed dielectrics.In (b), the touching point between the zones provides a pointliketransparent zone. In (c), the contact of two zones is a pointlikenontransparent gap. The phase deviations δϕ1,2 vanish on the reddashed lines.

gaps are degenerate in pointlike nontransparent zones (gaps)at the frequency ω = ωt . Note that the zone structure of thelattice composed of both left-handed dielectrics is similar tothe one shown in Fig. 4(a), but symmetrically reflected withrespect to the δk0 axis.

The band structure can be significantly different whenthe lattice contains both left- and right-handed dielectrics. Inthis case, there are transparency zones above and below thefrequency ωt = ck0t that touch one another at the frequencyω = ωt , forming a pointlike transparent zone as it is shown inFig. 4(b). This band structure is similar to the energy spectrumof the charge carriers in graphene, and manifests genuineDirac points. It must be emphasized that for such points toexist, the layers with both positive and negative refractiveindices must be present, but the average (over the period)value of n should not necessarily be equal to zero. Materialswith effective ε and/or effective μ near zero have becomerecently the subject of intensive investigation due to theirunusual transport properties, including the existence of conicalsingularities in the band gap structures,37,52,54–56 however, thistopic lies outside the domain of our paper.

It is worth noting that the zone structures shown in Fig. 4have been calculated for TE (p-polarized) waves. In the caseof s-polarized fields, the dielectric permittivities εi in Eq. (37)are replaced by the magnetic permeabilities μi , therefore thephotonic band structure is different, while the coordinates ofthe diabolic points are the same. This means that for a givenfrequency and angle of incidence, the same sample could betransparent for s polarization and opaque p polarization, andvice versa. This property can by utilized for the separation ofpolarizations.

The transparency zones depicted in Fig. 4 are the pro-jections onto the plane (ky,k0) of the surface ω = ω(kx,ky),described by the dispersion equation (33) and shown inFig. 5. The zone structures, presented in Figs. 4(a), 4(c),and 4(b), correspond to the surfaces shown in Figs. 5(a)and 5(b), respectively.

It is important to note that the presence of both left- andright-handed dielectrics in the periodic structure is a necessary,but not sufficient condition for the graphenelike band structureto exist. Generally speaking, the lines (41) and the bandedges do not coincide, and can be widely separated. Whenthe distance between the lines (41) and the zone edges is largeenough, as, for example, in Fig. 4(c), the zones shape is similar

FIG. 5. (Color online) Surfaces δk0(δk‖,δky) for (a) the zonestructure depicted in Fig. 4(b) and (b) the zone structures shownin Figs. 4(a) and 4(c).

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to that of a lattice formed by right-handed dielectrics only [seeFig. 4(c)]. A detailed analysis of the zones shape is presentedin Appendix.

The fundamental qualitative difference between the zoneshapes of mono- and mix-lattices owes its origin to the strongdispersion, which is an inherent characteristic of left-hadedmedia. Ignoring this fact (“for simplicity,” as this is sometimesdone) leads to wrong results for the zones structure in thevicinity of their contact.

Indeed, assuming a constant refractive index nj (k0) =const, the phase variation δϕj can be written as

δϕj = dj

kjx

(n2

j k0δk0 − kyδky

), (42)

i.e., the phase variation vanishes on the line that lies in the firstand the third quadrants, independently of the sign of nj .

B. Graphene superlattice

It follows from Eq. (39) that in a graphene superlattice cre-ated by an electrostatic potential with periodically alternatingvalues u1 and u2, the phase variations δϕ vanish on the lines

δw = h2v2F kyt

wt − uj

δky, (43)

where wt and kyt are the coordinates of the zone touching pointin the plane (ky, w). The slopes of these lines have oppositesigns when the energy wt lies between the values u1 and u2

of the potential, i.e., when sgn[(w − u1)(w − u2)] = −1. Inthis case, the zones touch each other either as it is shown inFigs. 4(b) or 4(c) (depending on the relation between the layerthicknesses d1 and d2, and the touching point indices p andq). In the opposite case, when sgn[(w − u1)(w − u2)] = +1,the zone structure is similar to the one shown in Fig. 4(a), orsymmetrically reflected with respect to the ordinate axis. Thusthe pointlike transparent zones (new Dirac points) can appearin the graphene superlattice only in the energy range betweenu1 and u2.

V. TRANSMISSION NEAR THE DIRAC POINTS

As it was mentioned in Introduction, the similarity betweenthe energy spectra of electromagnetic waves in homogeneousmedia and Dirac quasiparticles is only formal and physicallymeaningless, because the two cones in the photon spectra areidentical. To demonstrate that the singular points (presentedabove) in the band gap structure of mixed periodic dielectricsamples possess the properties that make them bona fide Diracpoints, we considered the transmission of a monochromaticwave of a frequency ω through a finite stack of alternatingleft- and right-handed dielectric slabs. The dependenciesof the amplitudes and phases on the complex transmissioncoefficients t(ky) of the y component of the wave vector [i.e.,of the angle of incidence θ = arcsin(cky/ω)] are shown inFigs. 6(a) and 6(b), for two different frequencies belonging,respectively, to the upper and the lower cones in Fig. 2. Whilethe amplitudes |t(ky)| are similar, the phases β = arg t(ky)manifest quite distinct behaviors:

β(ky) � ±b(ky − kym)2, (44)

FIG. 6. (Color online) The magnitude (blue) and the phase (red)of the complex transmission coefficient t(ky) of a monochromaticwave propagating through [(a) and (b)] finite periodic stacks ofalternating left- and right-handed dielectric layers and (c) a stackof normal dielectric layers. The frequencies in (a) and (b) belong tothe upper and lower cones in Fig. 2, respectively. The ky component ofthe wave vector determines the angle of incidence θ = arcsin (cky/ω).The parameters of the numerical simulations used here are number ofperiods N = 9, p = 1, q = 8, d1 = 0.35D, ε1 = n1 = 1, |ε2| = 0.8,|n2| = 2.5, |v2g| = 0.25c, and D|δk0| = 0.05.

where opposite signs correspond to two different cones, andkym is the position of the phase β(ky) extremum. For com-parison, the dependencies of |t(ky)| and β(ky) for a periodicstack of normal dielectric layers are shown in Fig. 6(c). In thiscase, the phase is a monotonic (close to linear) function of theangle of incidence at all frequencies and has no singularitiesat ky = kyt .

Exactly the same parabolic dependence of the phaseβ(ky) of the transmission coefficient is intrinsic to graphenesuperlattices, when the contact point corresponds to thepointlike transparent zone. The dependencies of the amplitudesand phases of the complex transmission coefficients t(ky) onthe y component of the wave vector are shown in Fig. 7for two different energies: one above [see Fig. 7(a)] andanother below [see Fig. 7(b)] the energy wt of the zonescontact.

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FIG. 7. (Color online) Magnitude (blue) and phase (red) ofthe complex transmission coefficient t(ky) of a monoenergeticquasiparticle propagating through [(a) and (b)] a finite periodic setof alternating p-n and n-p junctions in graphene. The quasiparticleenergies are (a) above and (b) below the energy of the zones touchingwt . The indexes of the zones contact point are p = 1, q = 5, andthe layer thicknesses are d1 = 0.3D and d2 = 0.7D. In the samelattice, the zones contact point, whose indexes are p = 3 and q = 1,corresponds to a pointlike gap, and the dependence t(ky) looks likethe one for a periodic stack of normal dielectrics [see Fig. 6(c)].

As it was mentioned above, the presence of both left- andright-handed dielectrics, or alternating p-n and n-p junctionsin graphene, is not sufficient for a point of zone touching tobe a pointlike transparency zone (Dirac point). In the samelattice, points of the zone- touching with different indexes p

and q can be either pointlike gaps, as shown in Fig. 7(c), orpointlike transparent zones, i.e., Dirac points [see Figs. 7(a)and 7(b)].

A number of interesting effects emerge in the vicinity ofthe Dirac point of the mixed periodic sample or graphenesuperlattice. Some of them are caused by the parabolicdependence (44) of the transmittance coefficient phase on theangle of incidence (of ky). Let us consider the propagationof a monochromatic beam of light, bounded in the transversedimension (Gaussian beam, for instance), through the mixedstack of a thickness L. Generally, a beam transmitted througha slab of a normal dielectric is shifted along the surface, as itis schematically shown in Fig. 8(a).

FIG. 8. (Color online) (a) A beam of light transmitted through aslab of a normal dielectric is shifted along the surface. The shift ofthe beam passing through a mixed periodic stack is much smaller,can be either positive or negative, or even equal to zero, as shownin (b).

The shift � is determined53 by evolution of the phase β ofthe transmission coefficient t :

� = − dβ(ky)

dky

∣∣∣∣ky=ky0

, (45)

where ky0 is the tangential component of the wave vector ofthe central ray in the incident beam. It is assumed in Eq. (45)that the angular width �ky of the incident beam is rathersmall. Since mixed periodic stacks are characterized by theparabolic dependence β(ky), Eq. (44), dβ(ky)/dky is small oreven equal to zero, in which case the longitudinal shift � isabsent completely.

The absence of the longitudinal shift, whendβ(ky)/dky |ky=ky0 = 0, can also be observed in the so-callednear-zero-index metamaterials57 and in 1D periodic latticeswith birefringent materials.58 It is important to note that inour case, this phenomenon has a different physical origin.

The parabolic dependence β(ky) affects also the curvatureof the transmitted beam phase front, i.e., it shifts the focalplane of the incident Gaussian beam. The value �f of thisshift is proportional to the thickness L of the mixed stack andis independent of the stack position on the beam trajectory.The sign of the shift �f depends on the sign of the detuningof the beam frequency ω from the Dirac-point frequency ωt .The focal plane is shifted forward (defocusing) when ω < ωt ,and backward (focusing) when ω > ωt .

This surprising focusing properties of mixed periodicsamples are demonstrated in Fig. 9. Of the two beams with thefrequencies resting on two different dispersive cones, the onecorresponding to the upper cone is focused by the sample, bluecurve, while the other (lower cone) is defocused, red line. Forcomparison, the black curve presents the intensity distributionin the beam propagated in free space. This phenomenon ishighly unusual by itself and also reinforces the similarityof the contact points of the cones to a genuine “optical”Dirac point: different cones are not identical, and representobjects with distinct physical properties—a sort of opticalparticle-antiparticle pair.

When the frequency ω of the incident beam coincides witha pointlike gap of the corresponding infinite structure, i.e., ω =ωt , one would expect an exponential decay of the transmittedbeam intensity Itr as a function of the mixed stack thickness L:Itr(L) ∝ exp(−γL). However, the well-pronounced constant

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FIG. 9. (Color online) Focusing properties of a mixed periodicsample. After the transmission through a mixed periodic sample (greyarea), the focus of the beam with a frequency belonging to the uppercone (blue curve) is shifted to the left with respect to the beampropagating in free space (black curve). The same sample shifts thefocus of the beam with frequency from the lower cone (red curve) inthe opposite direction. The parameters of the numerical simulationsused here are the number of periods N = 100, p = 2, q = 5,d1 = 0.4D, ε1 = n1 = 1, ε2 = −0.5, n2 = −0.5, v2g = −0.6c, andD|δk0| = 0.3.

asymptotic of the function L Itr(L) (see Fig. 10, red line),demonstrates an anomalously high intensity, decaying as 1/L.Such a diffusionlike dependence is one of the consequencesof the linear, Dirac conelike dispersion. In two-dimensionalphotonic crystals with triangular lattices of normal dielectricrods, this phenomenon was predicted in Ref. 19. In layeredstructures, it could take place only in the presence of left-handed elements.

Because of the similarity between the energy spectrumof relativistic electrons and the frequency band structure ofa mixed periodic dielectric structure, it is natural to assumethat the light beam propagation into 1mixed samples can beaccompanied by a Zitterbewegunglike phenomenon. Indeed,the spatial distribution of the energy flux inside the mixedsample manifests a trembling motion: the “center of gravity”of the flux oscillates in the transverse direction (along the y

axis). In Fig. 11, the spatial distribution of the energy fluxinside the mixed sample is shown. The oscillatory motion ofthe center of gravity Ic(y) is clearly seen in Fig. 12. Note thatFig. 11 is similar to Fig. 3 in Ref. 59, where the probabilityfunction of a moving electron (solution of the Dirac equation)is shown.

Figure 11 also demonstrates the two above-mentionedeffects: the absence of the longitudinal shift (the energypropagates normally to the sample boundary, whereas theangle of incidence of the beam is far from the normal) andthe focusing of the beam.

All these effects—the absence of the longitudinal shift, thefocusing of the beam, and the Zitterbewegung phenomenon—are also seen in graphene superlattices. As an example,the current density distribution in the Gaussian beam of

FIG. 10. (Color online) Intensity I (N ) (blue line) of the transmit-ted beam as a function of the distance of propagation inside a mixedstack. The distance is measured in the numbers of periods N . Thefrequency ω of the incident beam belongs to a pointlike transparencyzone (Dirac point) of the corresponding infinite structure, ω = ωt .The large-N asymptotic of the red line, NI (N ), is constant, whichmeans that the intensity is inversely proportional to the distance(diffusionlike dependence). The parameters of the numerical simula-tions here are p = 1, q = 8, d1 = 0.35D, ε1 = n1 = 1, ε2 = −0.8,n2 = −2.5, and v2g = −0.25c.

monoenergetic charge carriers propagating through a finitegraphene superlattice is depicted in Fig. 13.

Let us now compare the wave propagation through amixed dielectric slab or a graphene superlattice with thetransmission of an electromagnetic wave through a platemade of a homogeneous (right- or left-handed) dielectric.

FIG. 11. (Color online) Spatial distribution of the energy fluxof the beam propagating (from left to right) through the mixedsample. The angle of incidence �34◦. The boundaries of the sampleare marked by the vertical dashed lines. The “center of gravity” ofthe flux oscillates in the transverse (along the y axis) direction. Theparameters of the numerical simulations used here are the numberof periods N = 100, p = 4, q = 10, d1 = 0.4D, ε1 = n1 = 1, ε2 =−0.5, n2 = −1.5, v2g = −0.2c, Dδk0 = −0.07, and Dδky = 0.5.

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BALLISTIC CHARGE TRANSPORT IN GRAPHENE AND . . . PHYSICAL REVIEW B 87, 245134 (2013)

FIG. 12. Oscillations of the transverse coordinate of the “centerof gravity” of the beam propagating in the mixed sample.

In a small vicinity of the normal angle of incidence, thedependence β(ky) of the transmission coefficient phase β

on the tangential component of the wave number ky (thedependence on the angle of incidence) has the same parabolicform as discussed above [see Eq. (44)], with the only differencethat at homogeneous dielectric plates, kyt = 0 by definition:

β(ky) = ±bk2y. (46)

Here, the plus sign corresponds to right-handed dielectrics,while the minus sign corresponds to left-handed dielectrics.In the first case, the refraction is positive, i.e., the incidentbeam and the beam inside the plate lie on opposite sides fromthe normal, whereas in the second case, the beams lie on thesame side of the normal (negative refraction). Therefore the

FIG. 13. (Color online) Transmission of the Gaussian beam ofcharge carriers through a periodic set of p-n and n-p junctions ingraphene near a pointlike gap. The angle of incidence �48◦. Thesample boundaries are marked by the vertical dashed lines. Theparameters of the numerical simulations used here are the number ofperiods N = 100, p = 1, q = 5, d1 = 0.3D, u1 = −u2 = 20hvF /D,δw = −0.3hvF /D, and Dδky = 0.2.

shift � is positive for plates of a right-handed dielectric, andnegative for left-handed dielectrics. In other words, in normalsamples, ky and � have the same signs, while in metamaterialsthe signs are opposite. Amazingly, these two situations, eachinherent to different kinds of homogeneous materials, can beobserved in the same mixed periodically layered sample, andgraphene superlattice. Indeed, depending on which side ofthe Dirac point the frequency lies, the refraction is eitherpositive (upper cone) or negative (lower cone). In this regard,it is more appropriate to refer to these two cones as a“medium-antimedium” pair, rather than “particle-antiparticle,”as in homogeneous graphene.

VI. CONCLUSIONS

We have shown that some of the exotic properties of chargetransport in graphene can be reproduced in the propagation oflight through layered dielectric samples. Similarities and dis-tinctions between Maxwell and Dirac equations, and betweenthe corresponding boundary conditions have been studied.Although the equations for the real electric and magnetic fieldsare essentially different from those for the Dirac electrons,under some conditions they can be reduced to similar form.For example, Eq. (4) for the complex combinations given byEq. (3) coincides with the Dirac equations (2). Therewith,the role of the refractive index in graphene is played by thedifference between properly normalized values of the Fermienergy and the external electrostatic potential [see Eq. (5)].The boundary conditions for a Dirac quasiparticle incidenton a plane separating two areas with different potentials, andfor an electromagnetic wave propagating through an interfacebetween two layers of homogeneous dielectrics are, generallyspeaking, different. They coincide only when the impedancesof the layers are equal, and the direction of the propagationis normal to the boundary. It means that at normal incidence,any junction in graphene is analogous to a contact betweentwo perfectly matched dielectrics and therefore is absolutelytransparent to normally incident Dirac electrons. This providesa more intuitive insight into the physics of the Klein paradoxin graphene.

The analytical and numerical analysis of the photonic bandgap structures of infinite periodically layered systems revealsan infinite number of the so-called “diabolic points” (singularpoints of contact of two transparency zones) in the (ω,θ )spectral diagrams (examples are shown in Fig. 4). A distinctionneeds to be drawn between two types of these singularities:pointlike transparency zones, like in Figs. 4(a) and 4(c), andpointlike gaps in the spectrum, similar to the one presented inFig. 4(c). Although all three pictures in Fig. 4 are topologicallyequivalent, the transport properties of the corresponding finiteperiodic stacks of layers differ drastically in the vicinities ofthese points. Waves with frequencies lying on opposite sides ofthe singularities of the first type propagate through the samplesin similar ways. In the same time, when two touching spectralcones form a pointlike gap, the electromagnetic radiationinteracts with the same sample differently, depending to whichcone its frequency belongs. Studies of the propagation ofbeams of light show that only the diabolic points of this typeposses the properties of genuine Dirac points. We demonstratethat in monotype layered structures (i.e., in those built of either

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normal or left-handed dielectrics), just the diabolic point of thefirst type can exist, and conical, Dirac-type singularities appearonly in mixed (with alternating left- and right-handed layers)samples. This is in contrast to two-dimensional media whereDirac points were discovered in various types of photoniccrystals with normal dielectric elements. It is important to notethat the physical nature of the Dirac points that we consideris different from that in systems with zero average value ofthe refractive index: in mixed layered structures, they are dueto the specific strong dispersion (phase and group velocitieshave different signs) inherent in the elements with negativerefraction.

Although the angular dependencies of the transmissionand reflection coefficients from a single interface in layereddielectrics and graphene superlatices are different [compareformulas Eqs. (18) and (24)], the spectral properties of thesetwo structures are conceptually identical and entail similarfeatures in the light and charge transport. Considering, asexamples, the transmission of the Gaussian monochromaticbeams of light and monoenergetic Dirac electrons through thecorresponding (dielectric or graphene) samples we predict thefollowing Dirac-point-induced effects: (i) two touching Diraccones influence the propagation of a beam in different ways:the beam is focused when the frequency (energy) belongs tothe upper cone, and is defocused at frequencies (energies)lying in the lower one, (ii) the transverse shift of the beam isanomalously small or even zero, (iii) the decay of the intensityat forbidden frequencies is diffusionlike, and (iv) a spatialanalog of the Zitterbewegung effect (i.e., trembling motionof the “center of gravity” of the energy flux) is observed inperiodically layered dielectric structures with nonzero averagerefractive indices and in graphene superlattices.

ACKNOWLEDGMENTS

This work was supported in part by JSPS-RFBR Grant No.12-02-92100, RFBR Grant No. 11-02-00708, ARO, RIKENiTHES Project, MURI Center for Dynamic Magneto-Optics,Grant-in-Aid for Scientific Research (S), MEXT Kakenhi onQuantum Cybernetics, the JSPS via its FIRST program, andby the Israeli Science Foundation (Grant No. 894/10).

APPENDIX: BOUNDARIES OF THE TRANSPARENCYZONES OF PHOTONIC STRUCTURES

Boundaries of the transparency zones in the plane (δky,δk0)in the vicinity of the zones touching point are defined by theequation

δϕ21 + δϕ2

2 + 2aδϕ1δϕ2 = 0, (A1)

where δϕj are defined by Eq. (40). It follows from Eq. (A1),that

δϕ2 = (−a ±√

a2 − 1), (A2)

where

a = 1

2

(k1xε2

k2xε1+ k2xε1

k1xε2

).

This equation, using Eq. (40), can be presented in the form

δk(±)0 = δky

kyt

k0t

d2k1x − d1k2x(−a ± √a2 − 1)

d2k1xc

vg2− d1k2x

cvg1

(−a ± √a2 − 1)

. (A3)

When the photonic structure is formed by either right- or left-handed layers only, i.e, a > 0 and sgn(vg1/vg2) = 1, both linesδk

(±)0 (δky) lie in the same quadrants and the zones touching

point forms a pointlike gap, as it is shown, for instance, inFig. 4(a). In the mixed structure, which contains both left- andright-handed layers, i.e., a < 0 and sgn(vg1/vg2) = −1, thesituation is quite different. The denominator in Eq. (A3) hasthe same sign for both δk

(+)0 and δk

(−)0 , while the signs of the

numerator can be different. Whether the sign of (δk(+)0 /δk

(−)0 )

is equal to −1 or +1 depends on the values of the parameters.The zones structures in the first [sgn(δk(+)

0 /δk(−)0 ) = −1] and

the second [sgn(δk(+)0 /δk

(−)0 ) = +1] cases are similar to ones

shown in Figs. 4(b) and 4(c), respectively.Thus the zones touching point presents the pointlike

transparent zone when the following inequalities hold:

d2k1x

d1k2x

< |a| +√

a2 − 1,d2k1x

d1k2x

> |a| −√

a2 − 1. (A4)

Using the definition of the parameter a and Eq. (35) theseinequalities can be written as

(d1

d2

)3 (q

p

)2 (ε1

|ε2|)

< 1 <

(d1

d2

) ( |ε2|ε1

), (A5)

when

(d2

d1

) (p

q

) ( |ε2|ε1

)> 1,

and reverse to Eq. (A5) inequalities, when

(d2

d1

) (p

q

) ( |ε2|ε1

)< 1.

As it follows from Eq. (35), the range of allowed values of theparameters is restricted by the condition

(d2

d1

)(p

q

) ( |n2|n1

)> 1, (A6)

when n1 > |n2|, and by the reverse inequality when n1 < |n2|.All these inequalities select a region in the 4-dimensionalspace of parameters d1/d2, p/q, ε1/|ε2|, and n1/|n2|, wherethe vicinity of the zones contact point is characterized by theDiraclike spectrum.

Any metamaterial-dielectric pair is characterized by itsown values of the parameters ε1/|ε2| and n1/|n2|. Therefore,considering these parameters as given, one can define thecorresponding area in the two-dimensional space of parameters

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ξ = p/q and η = d1/d2 (remainder: p and q are integernumbers). Introducing the notations A = n1/|n2| and B =ε1/|ε2| we can write the inequalities Eqs. (A5) and (A6) as

Bη3 < ξ 2, η < B, Bη < ξ,(A7)

Bη3 > ξ 2, η > B, Bη > ξ,

and

ξ > Aη, if A > 1,(A8)

ξ < Aη, if A < 1.

Equation (A8) means that the line η = A−1ξ divides the plane(ξ,η) in two parts. The allowed values of variables ξ andη lie below this line if A > 1, and above the line if A < 1.Inequalities Eq. (A7) bound an area between the line η = B

and the curve η = (x2/B)1/3. The region where a touchingpoint of the zones presents the genuine Dirac point is definedby the intersection of these two areas, as it is shown in Fig. 14.

Note that the group velocities vg1 and |vg2| are not involvedin this analysis. These velocities define the angle of openingof the Dirac spectrum, but not the fact of its existence.

Usually, metamaterials exhibit their left-handed propertiesin a rather narrow frequency range. Because the above-mentioned region is defined only by the relation between thelayer thicknesses d1 and d2, one can tune the Dirac point

FIG. 14. Plot of the photonic structure parameters ξ = d1/d2 andη = p/q. The vicinity of the zones contact point is characterized by aDiraclike spectrum when the structure parameters belong in the grayregion below the dashed line (if A = n1/|n2| > 1) or above this line(if A = n1/|n2| < 1).

frequency ωt = ck0t by an appropriate choice of the structureperiod D.

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